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Definitions: A progression of numbers is called a sequence. E.g., 2, 4, 8, 16,…, 1024 The sequence can be finite if it ends or infinite if it is unending. In our example, the sequence ends with the 10th term, 1024, and is therefore a finite sequence. A term of a sequence (or series) is an individual number in the progression. In our example above, the 3rd term is 8. A Series is the sum of a progression of numbers. E.g., 2+4+8+…+1024. Often we are interested in the particular number that this sum equals. Ratios in a sequence are the numbers obtained by dividing a term by its previous term. In our example, the ratios are: 4/2=2 8/4=2 etc. An Geometric Sequence is a sequence in which all the ratios are the same. Our example 2, 4, 8, …, 1024 is a geometic sequence because the ratios all equal 2. A second example, 1, 3, 5, 7, …, 45 is not a geometic sequence because its ratios aren’t all the same. The General Term of a sequence is a kind of basket that represents all of the terms of the sequence at once. In our example 2, 4, 8, … the general term is {2x }. When x=3, we see that 23 = 8 is the third term. The general term makes it easy to predict that the 10th term is 210 = 1024 or that the 1034th term is 21034. Technique for Geometic Sequences 1. Find the ratio. This identifies the power (exponential) function that your sequence is a variation on, and hence tells you what base to raise to the x power. In the example 3, 6, 12, 24, … this ratio would be 2 telling us that this sequence is a variation on the 2x function: 2, 4, 8, 16,… 2. Find what term zero of the sequence would be. In our example above, term zero is what would come before the 3 in 3, 6, 12, 24,…. This would be 1.5. This is what must be multiplies by the power function of x found in step 1. 3. Put the above two parts together and you get the general term: {1.5(2x)}. 4. Now you are in a position to use the general term to make predictions. For instance, the 10th term is 1.5(210) = 1.5(1024) = 1536. We can also find out what term number is 192 by solving the equation 1.5(2x)=192 to find that 2x=128 and hence x=7. Find the 10th term of the sequence 20, 40, 80, … 10, Ratio: 2 Term 0: 5 General Term: 5(2x) Term 10: 5(210)=5(1024)=5120 Find the 20th term of the sequence 18, 54, 162, … Ratio: 3 Term 0: 2 General Term: 2(3x) 6, Term 20: 2(320)=2(3486784401)=6973568802 Find the 6th term of the sequence 200, 2000, … 20, Ratio: Term 0: General Term: Term 6: Find the 10th term of the sequence 9, 27, 81, … Ratio: Term 0: General Term: Term 10: 3, Find the 11th term of the sequence 24, 12, 6, … 48, Ratio: Term 0: General Term: Term 11: Find the 9th term of the sequence 576, 192, 64, … 1728, Ratio: Term 0: General Term: Term 9: Find the 10th term of the sequence 5, 25, 125, … Ratio: Term 0: General Term: Term 10: 1, Find the 8th term of the sequence 54, 36, 24, … Find the 13th term of the sequence 2/3, 4/3, 8/3, … 81, 1/3, Find the 10th term of the sequence 2/3, 4/9, 8/27, … 1, Find the 10th term of the sequence 1/2, 2, 8, … 1/8, Answers Page No. Example No. Ratio Term 0 General Term nth Term 3 1 2 5 5(2x) 5120 2 3 2 2(3x) 6973568802 1 10 2 2(10x) 2000000 4 5 6 2 3 1 3x 59049 3 1/2 96 96((1/2)x) 3/64 1 1/3 5184 5184((1/3) x) 64/243 2 5 1/5 (1/5)(5x) 1953125 3 2/3 1 2 1/6 (1/6)(2x) 4096/3 2 2/3 3/2 (2/3)((3/2)x) 512/19683 3 4 1/32 (1/32)(4x) 32768 243/2 (243/2)((2/3)x) 64/243 Green Valley School 389 Pembroke Street Pembroke, NH 03275 (603) 485-8550